Power system variables such as \(\mathbf {P}\) , \(\mathbf {Q}\) , \(\mathbf {V}\) , and \(\mathbf {\theta } \in \mathbb {R}^n\)  can be represented as graph signals by modeling the power grid as a graph.
Specifically, if we map buses to vertices \( (\) | = n\(), branches and transformers to edges (\) E\(), and line admittances to weighted adjacency matrix (\) W Rn n\(), we can efficiently represent the power grid topology with a connected, undirected, weighted graph \) G= ( E, W)\(.For \) G\(, the normalized graph Laplacian can be given by~(\ref {eq:laplacian}).\begin{equation}\mathbf {L} = \mathbf {I_n} - \mathbf {D}^{-1/2} \mathbf {W} \mathbf {D}^{-1/2} \in \mathbb {R}^{n \times n}\end{equation}The Laplacian, i.e. \) L\( is an essential operator in the theory of GSP.Since \) L\( is a real, symmetric, and positive semi-definite matrix, it can be factorized as \) L = U  UT\(where \) U = [u0, ..., un-1] Rn n\( denotes the \) n\( orthonormal eigenvectors, and \)  = *diag([0, ..., n-1]) Rn n\( corresponds \) n\( eigenvalues of \) G\(.Indeed, \) ui\( and \) i\( values form the Fourier basis and Fourier frequencies in spectral domain of \) G\( \cite {ortega2018graph}.\) Different from classical signal processing, there is no meaningful translation operation in the vertex domain [1]}.
Thus, to filter a vertex domain graph signal \(\mathbf {x} \in \mathbb {R}^{n}\)  with a filter \(g_\theta \)  defined on \(\mathcal {G}\) , \(\mathbf {x}\)  is first transformed into the spectral domain using Graph Fourier Transformation (GFT) by \(\tilde{\mathbf {x}} = \mathbf {U}^T \mathbf {x}\) .
Next, the spectral domain signal \(\tilde{\mathbf {x}} \in \mathbb {R}^{n}\)  is filtered in the spectral domain by \(\tilde{\mathbf {y}} = g_\theta (\mathbf {\Lambda }) \tilde{\mathbf {x}}\)  where \(g_\theta (\mathbf {\Lambda }) = \operatornamewithlimits{diag}(\mathbf {\theta })\)  is a filter kernel, and \(\mathbf {\theta } \in \mathbb {R}^n\)  is a vector of Fourier coefficients [2]}.
Finally, the result is transformed back to the vertex domain using inverse GFT by \(\mathbf {y} = \mathbf {U}^T \tilde{\mathbf {y}}\)  [1]}.
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